A firm produces a perishable food product at a cost of $10 per case. The product
ID: 395402 • Letter: A
Question
A firm produces a perishable food product at a cost of $10 per case. The product sells for $15 per case. For planning purposes the company is considering possible demands of 100, 200, or 300 cases. If the demand is less than production, the excess production is discarded. If demand is more than production, the firm in an attempt to maintain a good service image, will satisfy the excess demand with a special production run at a cost of $18 per case. The product, however, always sells at $15 per case.
a. Set up the payoff table for this problem.
b. If P(100) = 0.2, P(200) = 0.2, and P(300) = 0.6, should the company produce 100, 200, or 300 cases?
Explanation / Answer
Given:
Cost per unit = C = $10 per case
Selling price per unit = P = $15 per unit
Production volume options (Alternatives) = 100, 200, or 300 cases
Demand states (events) = 100, 200, or 300 cases
a.
If demand = production,
Payoff (Vij) = (price – cost) × demand = (15 – 10) x demand
Payoff (Vij) = $5 x demand
If demand < production, excess produced cases are discarded
Payoff (Vij) = price × demand – cost × Cases produced
Payoff (Vij) = [(15 – 10) × Cases sold] – (10 × Cases unsold)
Payoff (Vij) = (5 × Cases sold) – (10 × Cases unsold)
If demand > production, shortage volume is produced at rate of $18 per case (Selling price remains same)
Payoff (Vij) = [(Price – normal cost) × production volume] + [(price – special run cost) × (demand – production volume)
Payoff (Vij) = [(15 – 10) x production volume] + [(15 – 18) x (demand – production volume)]
Payoff (Vij) = [5 x production volume] + [-3 x (demand – production volume)]
Thus, payoff table is obtained as follows:
Demand
Production alternatives
100
200
300
100
(5*100)
= 500
(5*100)+[(-3)*(200-100)]
= 200
(5*100)+[(-3)*(300-100)]
= -100
200
(5*100) - (10*100)
= -500
(5*200)
= 1000
(5*200)+[(-3)*(300-200)]
= 700
300
(5*100) - (10*200)
= -1500
(5*100) - (10*200)
= 0
(5*300)
= 1500
b.
Expected payoff of alternative = sum of (probability x payoff)
Probability of Demand
0.2
0.2
0.6
Conditional Payoff
Expected payoff of alternative
Production/Demand
100
200
300
100
500
200
-100
(0.2*500)
= 100
(0.2*200)
= 40
(0.6*(-100))
= -60
(100+40+(-60))
= 80
200
-500
1000
700
-100
200
420
520
300
-1500
0
1500
-300
0
900
600
Maximum expected payoff is $600 for the alternative of producing 300 cases.
Demand
Production alternatives
100
200
300
100
(5*100)
= 500
(5*100)+[(-3)*(200-100)]
= 200
(5*100)+[(-3)*(300-100)]
= -100
200
(5*100) - (10*100)
= -500
(5*200)
= 1000
(5*200)+[(-3)*(300-200)]
= 700
300
(5*100) - (10*200)
= -1500
(5*100) - (10*200)
= 0
(5*300)
= 1500
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